Towards Real-Time Bayesian Inference for
Magnetoencephalography
Alberto Sorrentino, Adam M. Johansen, John A. D.
Aston, Tom Nichols, Wilfrid S. Kendall
University of Warwick
a.m.johansen@warwick.ac.uk
JSM: August 12th, 2015
1
Outline
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Background
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I
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Magnetoencephalography
Hidden Markov Models / State Space Models
Particle Filters / Sequential Monte Carlo
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Models for Filtering of MEG Data
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Examples
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Computational Considerations
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Conclusions
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Magnetoencephalography
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Noninvasive imaging technique which measures magnetic
fields induced by neural currents.
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Sampling rate ∼ 100Hz.
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Ill-posed inverse problem: turn measurements into
estimates of brain activity.
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Problem addressed: estimation of current brain activity
given observations received.
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Hidden Markov Models
x1
x2
x3
x4
x5
x6
y1
y2
y3
y4
y5
y6
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Unobserved Markov chain {Xn } transition f .
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Observed process {Yn } conditional density g.
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Density:
p(x1:n , y1:n ) = f1 (x1 )g(y1 |x1 )
n
Y
f (xi |xi−1 )g(yi |xi ).
i=2
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Formal Solutions
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Filtering and Prediction Recursions:
p(xn |y1:n−1 )g(yn |xn )
p(xn |y1:n ) = R
p(x0n |y1:n−1 )g(yn |x0n )dx0n
Z
p(xn+1 |y1:n ) = p(xn |y1:n )f (xn+1 |xn )dxn
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Smoothing:
p(x1:n |y1:n ) = R
p(x1:n−1 |y1:n−1 )f (xn |xn−1 )g(yn |xn )
g(yn |x0n )f (x0n |xn−1 )p(x0n−1 |y1:n−1 )dx0n−1:n
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Importance Sampling in This Setting
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Given p(x1:n |y1:n ) for n = 1, 2, . . . .
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Could sample from a sequence q1 (x1 ), . . . , qn (x1:n )
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If qn (x1:n ) = qn (xn |x1:n−1 )qn−1 (x1:n−1 )
we can re-use samples.
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Weight:
p(x1:n |y1:n )
qn (xn |x1:n−1 )qn−1 (x1:n−1 )
p(x1:n |y1:n )
=
wn−1 (x1:n−1 )
qn (xn |x1:n−1 )p(x1:n−1 |y1:n−1 )
f (xn |xn−1 )g(yn |xn )
wn−1 (x1:n−1 )
=
qn (xn |xn−1 )p(yn |y1:n−1 )
wn (x1:n ) ∝
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SIR Particle Filters [GSS93]
Can be viewed as an extension of importance sampling
I Algorithmically, at iteration n:
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I
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i
i
} for i = 1, . . . , N :
, X1:n−1
Given {Wn−1
ei
Resample, obtaining {1/N, X
1:n−1 }.
I
i
en−1
Sample Xni ∼ qn (·|X
)
I
Weight Wni ∝
i ei
i
f (Xn
|Xn−1 )g(yn |Xn
)
ei
qn (X i |X
)
n
n−1
Actually:
I
I
Resample efficiently.
Only resample when necessary.
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Iteration 2
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1
0
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Iteration 3
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Iteration 4
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Iteration 5
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Iteration 6
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Iteration 7
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Iteration 8
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Iteration 9
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Iteration 10
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Resample-Move Particle Filters [GB01]
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Originally, incorporate p(xt |y1:t )-invariant MCMC kernels.
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Here we consider p(x1:t |y1:t )-invariant kernels.
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Actually, go back only to the birth of the oldest surviving
dipole.
17
Measurement Model
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Work with discretized brain (Ngrid ≈ 10000 elements).
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Precompute ( Nsensors × 3Ngrid .) leadfield/gain matrix, G.
p(bt |jt ) = N
bt ;
Nt
X
G
(i)
rt
!
(i)
qt , Σnoise
i=1
where
jt =
(1)
(1)
rt , qt
(N ) (N )
, . . . , rt t , qt t
and G(r) is a Nsensors × 3 matrix for each r.
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A Random Walk with Births and Deaths
Mixture of 3 components:
p(jt |jt−1 ) =Pbirth (jt−1 )Tbirth (jt |jt−1 ))+
Pdeath (jt−1 )Tdeath (jt |jt−1 ))+
Prw (jt−1 )Trw (jt |jt−1 ))
Where:
(1:Nt−1 )
Tbirth (jt |jt ) =Trw (jt
Tdeath (jt |jt ) =
(Nt−1 +1)
|jt−1 )Ugrid (rt
(Nt−1 +1)
)N (qt
; 0, σq I)
Nt−1
1
X
Nt−1
(i)
(i)
Trw (jt |jt−1 \ (rt−1 , qt−1 ))
i=1
Nt−1
Trw (jt |jt ) =
Y
(i)
(i)
(i)
(i)
N |grid (rt ; rt−1 )N (qt ; qt−1 , ∆)
i=1
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A Static Dipole Model [SJA+ 13]
Mixture of 3 components:
p(jt |jt−1 ) =Pbirth (jt−1 )Tbirth (jt |jt−1 ))+
Pdeath (jt−1 )Tdeath (jt |jt−1 ))+
Psd (jt−1 )Tsd (jt |jt−1 ))
Where:
(1:Nt−1 )
Tbirth (jt |jt ) =Tsd (jt
Tdeath (jt |jt ) =
(Nt−1 +1)
|jt−1 )Ugrid (rt
(Nt−1 +1)
)N (qt
; 0, σq I)
Nt−1
1
X
Nt−1
(i)
(i)
Tsd (jt |jt−1 \ (rt−1 , qt−1 ))
i=1
Nt−1
Tsd (jt |jt ) =
Y
i=1
(i)
(i)
(i)
δr(i) (rt )N (qt ; qt−1 , ∆)
t−1
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Simulation Experiment
30
S1
S2
S3
S4
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Strength
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15
10
5
0
0
50
100
150
100
150
Time [ms]
−11
1
x 10
Amplitudes
0.5
0
−0.5
−1
−1.5
0
50
Time
Source locations (top row S1 and S2, bottom row S3 and S4),
source time courses and generated noisy field.
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Static
RandomWalk
dSPM
Music
L1L2
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Application to Real Data
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Somatosensory Evoked Fields mapping experiment
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Acquired with 306-channel MEG device [204 planar
gradiometers; 102 magnetometers]
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The left median nerve at wrist was electrically stimulated
at the motor threshold (interstimulus interval randomly
varying between 7.0 s and 9.0 s).
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Signals filtered to 0.1-200 Hz and sampled at 1000 Hz.
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Electrooculogram (EOG) used to monitor eye movements.
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Trials with EOG or MEG exceeding 150 mV or 3 pT/cm
were excluded.
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84 clean trials were averaged.
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To reduce external interference, signal space separation
method was applied to the average.
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A 3D digitizer and four head position indicator coils were
employed to determine the position of the subject’s head.
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Static
RandomWalk
dSPM
Music
L1L2
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Computational Considerations: A Related PET
Algorithm [ZJA15]
Runtime of algorithms
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Variance of estimates
0.32
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0.16
Implementation
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0.08
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MCMC−CPU1
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SMC−CPU1
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SMC−CPU4
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SMC−GPU
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0.04
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0.01
0.125
0.25
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32
Wall clock time (sec)
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Conclusions
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Modelling and computational considerations interact.
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Static dipole models allow better interpretation.
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Online inference under such models is possible with care.
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Computational costs are manageable. . .
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and parallelisation is easy.
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References
W. R. Gilks and C. Berzuini.
Following a moving target – Monte Carlo inference for dynamic Bayesian
models.
Journal of the Royal Statistical Society B, 63:127–146, 2001.
N. J. Gordon, S. J. Salmond, and A. F. M. Smith.
Novel approach to nonlinear/non-Gaussian Bayesian state estimation.
IEE Proceedings-F, 140(2):107–113, April 1993.
A. Sorrentino, A. M. Johansen, J. A. D. Aston, T. E. Nichols, and W. S.
Kendall.
Filtering of dynamic and static dipoles in magnetoencephalography.
Annals of Applied Statistics, 7(2):955–988, 2013.
Y. Zhou, A. M. Johansen, and J. A. D. Aston.
Towards automatic model comparison: An adaptive sequential Monte Carlo
approach.
Journal of Computational and Graphical Statistics, 2015.
In press.
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